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Contextual Symbols in Math

In my book I discuss the importance of context in reading and writing mathematics. An early step in becoming comfortable with math is deciphering the syntax of mathematical expressions. Another is in connecting the symbols to their semantic meanings. Embedded in these is the subproblem of knowing what to call the commonly used symbols. The more abstract you go, the more exotic the symbols tend to get.

Wikipedia has an excellent list for deciphering those symbols that have a typically well-understood meaning, like and . There is another list for common associations of Greek letters in science and math, along with the corresponding English/Latin list. There’s also a great website called Detexify that guesses the name of a symbol from your drawing. It’s a great way to lookup a confusing symbol.

To augment these resources, I’ll describe a few context-clues I’ve picked up over the years, and my first-instinct contextual association of each Greek letter. I wonder if there should be a database of such contextual associations.

Context clues

Variables represent a word with a related starting letter or sound. E.g., for “function,” or for “time.” Greek letters do this too. For example, (lower case pi) is the Greek “p”, and it might be used for a “projection” function. Or $\latex Gamma$ (capital gamma, the Greek “G”) for a “graph.” This can help, for example, when trying to determine the type of the variable . In many cases you can quickly deduce it’s a vector.

Capital letters and lower case letters are usually related. For example, might be a member of the set . A function might be constructed from another function in such a way that all the information in is preserved, but is somehow “bigger” (e.g., the relationship between the probability density function and the cumulative density function). For this reason, it can help to know greek counterparts, e.g., that is the Greek lower case of $\latex Sigma$.

Adjacent letters are often related, both withinin and across alphabets. The variables are often used together to represent different parts of the same object, while letters like are used to represent a semantically different part of the object. For example, carries a strong association that are fixed constants and are unknown variables. Greek does this too. A triangle might have its side-lengths as , and for each side length the opposite angle to that side gets the corresponding first three Greek letters .

Fonts can imply semantics. The blackboard-bold font represents systems of numbers, as in . The lower-case fraktur font represents ideals in ring theory, particularly prime ideals, like . Calligraphic fonts like are used for higher-order structures after the context of lower-case and capital letters are already set, like categories (calligraphic) of sets (capital) of elements (lower case). I have seen some cases where sans-serif fonts are used in this role when calligraphic fonts are taken.

I associate lowercase gamma with the Euler-Mascheroni constant, and nu with neutrinos. For permutations, the standard is lowercase sigma; lowercase pi also gets used for this. For specific functions, we have pi for the prime-counting function, capital Lambda for the Liouville function, and lowercase mu is the Möbius function. Lowercase tau is also sometimes used for 2pi. Capital and lowercase omega represent the number of prime factors of an integer with and without multiplicity, respectively.

There are tons of these contexts in theoretical and applied math&engineering, and they all use things slightly differently. Several years ago, someone tried to make a “formula text search” engine, where you would type in parts of a formula that you remember from school, and it would find the right formula and give you the complete context. It was a valiant effort, but back then the search tools where far too primitive. It is probably possible with today’s deep learning-based NLP technology: GPT-2, Bert, etc..

1. Delta: diagonal of a cartesian product, i.e. Delta(A)={(a,a) for a in A} is a subset of the cartsian product AxA.

2. Omega: an open connected set of an euclidean space, most often the complex plane; Or, in a completely unrelated use, Omega can be a probability space (the set of possible outcomes for a random experiment).